Peter Cameron question
Prof. Peter Cameron has put a quick question in his blog:
for which positive integers n are \(2^n1\) and \(2^n+1\) both products of at most 2 distinct primes? Link: [URL]https://cameroncounts.wordpress.com/2020/10/07/betweenfermatandmersenne/[/URL] 
87 Mersenne semi primes known so far.
The last 23 from M86371 to M10443557 are "semi probable primes" or "probable semi primes" (biggest factors are PRP). [url]https://www.mersenne.ca/prp.php[/url] [CODE] M11 M23 M37 M41 M59 M67 M83 M97 M101 M103 M109 M131 M137 M139 M149 M167 M197 M199 M227 M241 M269 M271 M281 M293 M347 M373 M379 M421 M457 M487 M523 M727 M809 M881 M971 M983 M997 M1061 M1063 M1427 M1487 M1637 M1657 M2357 M2927 M3079 M3259 M3359 M4111 M4243 M4729 M5689 M6043 M6679 M7331 M7757 M10169 M14561 M17029 M26903 M28759 M28771 M58199 M63703 M86371 M106391 M130439 M136883 M151013 M173867 M221509 M271211 M271549 M406583 M432457 M611999 M684127 M1010623 M1168183 M1304983 M1629469 M2327417 M3464473 M4187251 M5240707 M7313983 M10443557 [/CODE] 
We can only make an unbroken and minimal list of integers exclusively less than 1277, since we don't know if M1277 is a semiprime or not.
ETA As far as the Mersenne number candidates are concerned any nonsemiprine exponent is sieved out. ETA ll Or is it nonprime exponents? 
We are looking at n prime. Since 2^n+1 has a forced factor of 3, we're looking for Wagstaff primes where the corresponding Mersenne number is a prime or semiprime. There should be infinitely many Wagstaff primes. My guess is infinitely many cases where corresponding Mersenne number is a semiprime.

[QUOTE=a1call;560151]We can only make an unbroken and minimal list of integers exclusively less than 1277, since we don't know if M1277 is a semiprime or not.[/QUOTE]
Is 2^1277+1 a semiprime? You don't need the status of both 1 and +1 forms if one side is ruled out, the other side doesn't matter. 
[QUOTE=VBCurtis;560156]Is 2^1277+1 a semiprime? You don't need the status of both 1 and +1 forms if one side is ruled out, the other side doesn't matter.[/QUOTE]
I should stop posting in my sleep. Please disregard my post. (3, 888793, 3432577, ... )  F1277 
If I read the merssene.org correctly, M5807 is a (Probable) semiprime and so is F5807.
If I read the merssene.org correctly, M14479 is a (Probable) semiprime and so is F14479. If I read the merssene.org correctly, M42737 is a (Probable) semiprime and so is F42737. So no practical upper bound for a continuous list below them.:smile: 
[QUOTE=a1call;560160]If I read the merssene.org correctly, M5807 is a (Probable) semiprime and so is F5807.
If I read the merssene.org correctly, M14479 is a (Probable) semiprime and so is F14479. If I read the merssene.org correctly, M42737 is a (Probable) semiprime and so is F42737. So no practical upper bound for a continuous list below them.:smile:[/QUOTE] If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.:smile: 
[QUOTE=a1call;560161]If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.:smile:[/QUOTE]
I appreciate your work to list these, and I think I'll have a go at some ECM on M117239 'cause this is fun. 
Glad to have been of service if any.
Otherwise I'm just glad to have something to do with my new build.:smile: 
[QUOTE=a1call;560160]If I read the merssene.org correctly, M5807 is a (Probable) semiprime and so is F5807.
If I read the merssene.org correctly, M14479 is a (Probable) semiprime and so is F14479. If I read the merssene.org correctly, M42737 is a (Probable) semiprime and so is F42737. So no practical upper bound for a continuous list below them.:smile:[/QUOTE] No, all those Mersenne numbers you listed have one factor and a (definitely) composite cofactor, so at least three factors. 
[QUOTE=Viliam Furik;560171]No, all those Mersenne numbers you listed have one factor and a (definitely) composite cofactor, so at least three factors.[/QUOTE]
I see, thanks for the explanation.:smile: "PRP Cofactor" title is probably not the best choice of words then. 
[QUOTE=a1call;560161]If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.:smile:[/QUOTE]
A small notational quibble: doesn't [TEX]F_n[/TEX] typically denote the nth Fermat number, i.e. [TEX]2^{2^n}+1[/TEX]? 
I'm sure it does. I never hid the fact that I am no mathematician.
Thank you for the correction. I will try to remember the notation.:smile: 
[QUOTE=a1call;560173]I see, thanks for the explanation.:smile:
"PRP Cofactor" title is probably not the best choice of words then.[/QUOTE] It is, because if cofactor (the rest of the number, when divided by known certainlyprime factors) is PRP, then if the number has only one certainlyprime factor, it means it is most probably a semiprime. If the cofactor is tested composite by PRP test, the cofactor is certainly composite. If you refer to [URL="https://www.mersenne.org/report_exponent/?exp_lo=5807&exp_hi=&full=1"]this page[/URL], then yes, it may be kind of confusing to see PRP cofactor written there, but that's really only a title. If it would be PRP, for real, it would have had written "PRP_PRP_PRP_PRP_" in place of residue. 
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